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The gas transmission problem when themerchant and the transportation functions
are disconnected.
Bouchra Bakhouya,Ieseg, Universite Catholique de Lille,3 rue de la digue, 59.800 Lille, France,
e-mail: [email protected]
Daniel De Wolf,Institut des Mers du Nord HEC Ecole de Gestion de l’ULG
Universite du Littoral Cote d’Opale Universite de Liege
49 Place d de Gaulle, BP 5529, Boulevard du Rectorat 7 (B31)
59 383 Dunkerque Cedex 1, France 4000 Liege, Belgium
e-mail: [email protected] e-mail: [email protected]
January 16th, 2007
Abstract
In [7], De Wolf and Smeers consider the problem of the gas distri-bution through a network of pipelines. The problem was formulated asa cost minimization subject to nonlinear flow-pressure relations, ma-terial balances and pressure bounds. The practical application was theBelgian gas transmission network.
This model does not reflect any more the current situation on thegas market. Today, the transportation and gas buying functions areseparated. This work considers the new situation for the transportationcompany. The objective for the transportation company is to determinethe flows in the network that minimize the energy used for the gas
1
transport. This corresponds to the minimization of the power used inthe compressors.
1 Introduction
In De Wolf and Smeers [7], the objective for the transportation companyis to satisfy the demand at several points of the network by buying gas atminimal prize. Today, the transportation function and gas buying functionare separated. For example, on the Belgian gas market, the transport isdevoted to Fluxys company. On the other side, several actors are in chargeof gas supplying (such as Distrigaz, the former integrated company).
In order to reflect this new situation, a modelization of the compressorsis introduced in the model of De Wolf and Smeers. The introduction ofthe compressors change the nature of the optimization problem. In fact,the problem considered by De Wolf and Smeers [7] was a non convex butseparable non linear problem. This problem was solved using successivepiecewise linear approximations of the problem.
In the present case, the relation between the pressure increase and theflow in the compressor is non separable. See, for example, Jean Andre et al[3], Babu et al [5] or Seugwon et al [10].
This non linear non convex problem is solved using a preliminary problem,namely the problem suggested by Maugis [9], which is a convex problem easyto solve. We show, on the example of the Belgian gas network that thisauxiliary problem gives a very good starting point to solve the highly nonlinear complete problem.
Future researches are devoted to introduce this model in a dimensioningmodel such as [6]. In this case, the gas transmission problem is the secondstage problem of a two stage problem, the first stage being the pipes andcompressors dimensioning problem.
The paper is organized as follows. The formulation of the problem ispresented in section 2, the solution method being discussed next. Section 4introduces two test problems based on the Belgian gas transmission system.This section also illustrates the utility of the first problem to find a goodstarting point. Conclusions terminate the paper.
2
2 Formulation of the problem
The formulation of the problem presented in this section applies thus to asituation where the gas merchant and transmission functions are separated.The transportation company must decide the gas flow in each pipe and thelevel of compression in each compressor to satisfy fixed demands distributedover different nodes at some minimal guaranteed pressure, the income of gasbeing also given. For the supply, we have preserved a flexibility close thatallows to take gas between 90 % and 110 % of the daily nominal quantity.
The network of a gas transportation company (See Figure 1) consists ofseveral supply points where the gas is injected into the system, several de-mand points where gas flows out of the system and other intermediate nodeswhere the gas is simply rerouted. Pipelines or compressors are representedby arcs linking the nodes.
The following mathematical notation is used. The network is defined asthe pair (N, A) where N is the set of nodes and A ⊆ N ×N is the set of arcsconnecting these nodes. The network can be represented as in Figure 1.
sg
si
sk
sjg
i
k
jpi pj
h l o
fij
dh = −sh dl = −sl do = −so
Figure 1: Network representation
Since we have preserved the flexibility close, two variables are associatedto each node i of the network: pi represents the gas pressure at this nodeand si the net gas supply in node i. A positive si corresponds to a supply ofgas at node i. A negative si implies a gas demand di = −si at node i.
3
A gas flow fij is associated with each arc (i, j) from i to j. There are twotypes of arcs: passive arcs (whose set is noted Ap) correspond to pipelinesand active arcs correspond to compressors (whose set is noted Aa).
The constraints of the model are as follows. At a supply node i, the gasinflow si must remain within take limitations specified in the contracts. A gascontract specifies an average daily quantity to be taken by the transmissioncompany from the producer. Depending on the flexibility of the contract,the transmission company has the possibility of lifting a quantity rangingbetween a lower and an upper fraction (e.g. between 0.90 and 1.1) of theaverage contracted quantity. Mathematically:
si ≤ si ≤ si
At a demand node, the gas outflow −si must be greater or equal to di, thedemand at this node.
The gas transmission company cannot receive gas at a pressure higherthan the one insured by the supplier at the entry point. Conversely, at eachexit point, the demand must be satisfied at a minimal pressure guaranteedto the industrial user or to the local distribution company. Mathematically:
pi ≤ pi ≤ pi
The flow conservation equation at node i (see Figure 2 for the flow conser-vation at a supply node i) insures the gas balance at node i. Mathematically:
∑j|(i,j)∈A
fij =∑
j|(j,i)∈A
fji + si
si
i fijfji
Figure 2: Supply node i
Now, consider the constraints on the arcs. We distinguish between thepassive and active arcs. For a passive arc, the relation between the flow fij
4
in the arc (i, j) and the pressures at the two ends of the pipe pi and pj is ofthe following form (see O’Neill and al.[11]):
sign(fij)f2ij = C2
ij(p2i − p2
j), ∀(i, j) ∈ Ap
where Cij is a constant which depends on the length, the diameter and theabsolute rugosity of pipe and on the gas composition. Note that the pipesflow variables fij are unrestricted in sign. If fij < 0, the flow −fij goes fromnode j to node i.
For an active arc corresponding to a compressor, the following expressionof the power used by the compressor can be found, for example, in Andre etal [3], Babu et al [5] or Seugwon et al [10]):
Wij =1
kηad
100
3600
P0
T0
TiZm
Z0
γ
γ − 1fij
(
pj
pi
) γ−1γ
− 1
(1)
with the following parameters:P0 = 1.01325 bar,T0 = 273.15 K,γ = 1.309,Z0 = 1,k = 0, 95 × 0, 97 × 0, 98,Zm = mean compressibility factor of the gas,ηad = the efficacity factor,Ti = the gas temperature at the entry of the compressor.
Note also that in (1), the gas flow is expressed in m3 per hour, the dissi-pated power in kW, the pressures pi and pj are in bar. Using mean values fortheses factors (Zm = 0,9, Ti = 288,15 K, ηad = 0.75 for a turbo-compressor,0.8 for a moto-compressor), the energy dissipated can be written as:
Wij = γ1fij
((pj
pi
)γ2
− 1
)(2)
with γ1 = 0.167 for a turbo-compressor, 0.157 for a moto-compressor and γ2
= 0.236.The power used in the compressor must be lower than the maximal power,
noted Wij:Wij ≤ Wij
5
There is also an upper bound on the maximal pressure increase ratio:
pj
pi
≤ γ3
The constant γ3 = 1.6 in most of practical cases.For active arcs, the direction of the flow is fixed:
fij ≥ 0, ∀(i, j) ∈ Aa
The objective function of the gas transmission company is to minimizethe energy used in the compressor. This can be written:
min z = α∑
(i,j)∈Aa
1
0, 9ηtherm
Wij (3)
where α is the unitary energy price (in Keuro/kW) and ηtherm is the thermicefficacity of the compression station.
The gas transmission problem can thus be formulated as follows:
min z(f, s, p, W ) = α∑
(i,j)∈Aa
1
0, 9ηtherm
Wij
s.t.
∑j|(i,j)∈A
fij =∑
j|(j,i)∈A
fji + si ∀i ∈ N (4.1)
sign(fij)f2ij = C2
ij(p2i − p2
j) ∀(i, j) ∈ Ap (4.2)
Wij = γ1fij
((pj
pi
)γ2
− 1
)∀(i, j) ∈ Aa (4.3)
si ≤ si ≤ si ∀i ∈ N (4.4)
pi ≤ pi ≤ pi ∀i ∈ N (4.5)
fij ≥ 0 ∀(i, j) ∈ Aa (4.6)pj
pi
≤ 1.6 ∀(i, j) ∈ Aa (4.7)
Wij ≤ Wij ∀(i, j) ∈ Aa (4.8)
(4)
It can be seen that the problem (4) is separable for equation (4.2) but notseparable for equation (4.3).
6
3 Solution Procedure
We now consider the solution of the problem (4). As already seen in De Wolfand Smeers [7], the formulation of the gas flow-pressure relation in pipes (4.2)is clearly nonconvex. The problem without the compressors modelization(4.3) was separable allowing to solve the problem by successive piecewiselinear approximations of the problem. Now the compressors modelization(4.3) renders the problem non separable.
Some procedure is thus required in general to tackle the nonconvexity ofthe problem, if only to find a local solution. The approach proposed here isto proceed by successively solving two problems, the first one being expectedto produce a good initial point for the second one.
Convergence in nonlinear programming may indeed crucially depend ona good choice of the starting point and this is especially true when the prob-lem is nonconvex. Our first problem is obtained by relaxing the pressureconstraints and eliminating all compressors in the full model.
The solution of this problem is conjectured to provide a good startingpoint for the second problem which is the complete model with pressurebounds and compressors. The same procedure was already proposed forsolving the problem of an integrated transmission and merchant gas companyby De Wolf and Smeers [7]. But here, we shall show on examples that thesolution of first problem is a very good starting solution for the completeproblem. In fact, we shall prove that they have similar objective functions(namely to minimize the energy dissipated in the network to transport thegas).
3.1 First problem: find a good initial point.
Consider the following convex problem which only accounts for pressure lossesalong the pipelines:
min h(f, s) =∑
(i,j)∈A
|fij|f 2ij
3C2ij
s.t.∑
j|(i,j)∈A
fij −∑
j|(j,i)∈A
fji = si ∀i ∈ N
si ≤ si ≤ si ∀i ∈ N
(5)
7
Since the problem is strictly convex in the flow variables, its optimalsolution is unique. The first constraints of (5) insure that the solution is alsounique in the supply variables. It is easy to prove the following results forproblem (5):
Proposition 1 The unique optimal solution of the problem (5) satisfies thenonlinear flow pressure relation (4.2).
Proof: Let πi be the dual variable associated to the gas balance constraintat node i. The Karush-Kuhn-Tucker necessary conditions (See Bertsekas [4,page 284]) satisfied at the optimum solution of the problem (5) can be writtenas:
sign(fij)f 2
ij
C2ij
= πi − πj ∀(i, j) ∈ A
This is exactly the nonlinear flow pressure relation (4.2) if we define:
πi = p2i , ∀i ∈ N
As remark in De Wolf and Smeers [7], there is no sign constraint on theπ variables since πi is the Lagrange multiplier associated to an equality con-straint, namely the gas balance equation at node i. Thus directly replacingπ = p2
i is not allowed. Let π be the value of the lowest dual variable:
π = mini∈N
{πi}
Define nowp2
i = πi − π.
We obtain exactly the flow pressure relations (4.2). �
The optimal solution of (5) satisfies thus all the constraints of (4) exceptthe pressure bounds constraints (4.5) and the compressors modelization (4.3),(4.7) and (4.8).
It can be shown (See De Wolf and Smeers [8]) that problem (5) hasa physical interpretation. Namely, its objective function is the mechanicalenergy dissipated per unit of time in the pipes. This implies that the pointobtained by minimizing the mechanical energy dissipated in the pipes shouldconstitute a good starting point for the complete problem.
Definition 1 The mechanical energy is defined as the energy necessary forcompressing fij from pressure pi to pressure pj.
8
Extending the work of Maugis for distribution network, we have the fol-lowing proposition.
Proposition 2 The objective function of problem (5) corresponds up to amultiplicative constant to the mechanical energy dissipated per unit of timein the pipes
Proof: At node i, the power Wi given by a volumetric outflow of Qi unitsof gas per second at pressure pi can be calculated in the following manner.The total energy released by the gas when changing pressure from pi to thereference pressure po is:
Wi =∫ pi
p0
Q(p)dp
By using the perfect gas state relation (p0Q0 = pQ), we can write:
Wi =∫ pi
p0p0Q0dp
p= p0Q0log(
pi
p0)
Denote the volumetric flow going though arc (i, j) under standard condi-tions by Q0
ij and the pressures at the two ends of the arc by pi and pj. Thepower lost in arc (i, j) can be calculated by:
Wij = Wi − Wj = Q0ijp0log(
pi
pj
) = Q0ij
p0
2log(
p2i
p2j
)
This power loss can be expressed through the head discharge variable Hij =p2
i − p2j as follows. Introducing the mean of square of pressure pM defined by
p2M =
p2i + p2
j
2,
the power discharge can be rewritten as
Wij = Q0ij
p0
2log(
p2M + Hij
2
p2M − Hij
2
)
= Q0ij
p0
2
[log(1 +
Hij
2p2M
) − log(1 − Hij
2p2M
)
]
9
Note that since Hij is small with respect to p2M , we obtain the following first
order approximation
Wij ≈ Q0ijp
0 Hij
2p2M
= Q0ijp
0p2i − p2
j
2p2M
The power discharge through the whole network is thus (we suppose that pM
is similar for each arc (i, j) and can be factored out in the following sum):
W ≈ p0
2p2M
∑(i,j)∈A
Q0ij(p
2i − p2
j) =p0
2p2M
∑(i,j)∈A
(Q0ij)
3
C2ij
sign(Q0ij)
which corresponds, up to a multiplicative constant, to the first term of the ob-jective of problem (5). We can thus conclude that the function h corresponds,up to a multiplicative constant, to the mechanical energy dissipated perunit time in the network due to the flow of gas in the pipes. �
This proposition was suggested to us by Mr. Zarea, direction de larecherche of Gaz de France.
Problem (5), being a convex separable problem, can be solved by anyconvex non linear optimization method. We have solve this problem usingGAMS/CONOPT. The resolution with GAMS leads to a solution (f ∗, s∗)with associated Lagrange multipliers π∗ which are used as starting point forthe complete problem.
3.2 Second problem: the complete problem.
The solution to the first problem satisfies all the constraints of problem (4)except for the pressure bounds and the compressors modelization equations.We now return to problem (4) and solve it using also GAMS/CONOPT. Aswe shall see in the following section, the flow variables founded in the firstproblem are the (local) optimal values of the complete problem. Thus thefirst problem give a very good starting point for the second problem.
This is not a surprizing since the objective function of problem (5) isthe mechanical energy dissipated in all the network since the objective of thecomplete problem (4) is to minimize the energy dissipated in the compressors.The only task for CONOPT for solving the complete problem (4) from thesolution of problem (5) is to restore the bounds on pressure.
10
4 Application to the Belgium gas network
To illustrate the utility of the first problem (5), we apply our suggestedsolution procedure to the schematic description of the Belgium gas networkgiven in De Wolf and Smeers [7] (See figure 3).
Algerian�gas
From storage
Norvegian gas
To France
To Luxemburg
Zeebrugge
Dudzele
BruggeZomergem
Gent
Antwerpen
Loenhout
Poppel Low�gasHigh�gas
Hasselt
Brussel
Mons
Blaregnies
PeronnesAnderlues
Warnand-Dreye
Namur
Wanze
Sinsin
Arlon
Berneau
Liege
From storageDutch�gas Node
Compressor
Voeren
From storage
Figure 3: Schematic Belgium gas network
Recall that Belgium has no domestic gas resources and imports all its nat-ural gas from the Netherlands, from Algeria and from Norway. The Belgiumgas transmission network carries two types of gas and is therefore dividedin two parts. The high calorific gas (10 000 kilocalories per cubic meter),comes from Algeria and Norway. The gas from Algeria comes in LNG format the Zeebrugge terminal and the gas from Norway is piped through theNetherlands and crosses the Belgian border at Voeren (See Figure 3). Thegas coming from the Netherlands is a low calorific gas (8 400 kilocalories percubic meter). We consider only to the high calorific network.
11
The reader is referred to De Wolf and Smeers [7] for a more detailed de-scription of the Belgium network. We summarize all the relevant informationconcerning the nodes in Table 1.
node town si si pi pi
106 scm 106 scm bar bar1 Zeebrugge 8.870 11.594 0.0 77.02 Dudzele 0.0 8.4 0.0 77.03 Brugge -∞ -3.918 30.0 80.04 Zomergem 0.0 0.0 0.0 80.05 Loenhout 0.0 4.8 0.0 77.06 Antwerpen -∞ -4.034 30.0 80.07 Gent -∞ -5.256 30.0 80.08 Voeren 20.344 22.012 50.0 50.09 Berneau 0.0 0.0 0.0 66.2
10 Liege -∞ -6.365 30.0 66.211 Warnand 0.0 0.0 0.0 66.212 Namur -∞ -2.120 0.0 66.213 Anderlues 0.0 1.2 0.0 66.214 Peronnes 0.0 0.96 0.0 66.215 Mons -∞ -6.848 0.0 66.216 Blaregnies -∞ -15.616 50.0 66.217 Wanze 0.0 0.0 0.0 66.218 Sinsin 0.0 0.0 0.0 63.019 Arlon -∞ -0.222 0.0 66.220 Petange -∞ -1.919 25.0 66.2
Table 1: Nodes description
For each compressor, the maximal power is given in in Table 2. Note that
Localization Wij (kW)
Berneau 20 888
Sinsin 3 356
Table 2: Compressors description
the compressor in Berneau corresponds to a large compression station.
12
For each pipeline in the network, the length, interior diameter and corre-sponding C2
ij are given in Table 3.
arc from to diameter length C2ij
[mm] [km]1 Zeebrugge Dudzele 890.0 4.0 9.070272 Zeebrugge Dudzele 890.0 4.0 9.070273 Dudzele Brugge 890.0 6.0 6.046854 Dudzele Brugge 890.0 6.0 6.046855 Brugge Zomergem 890.0 26.0 1.395436 Loenhout Antwerpen 590.1 43.0 0.1002567 Antwerpen Gent 590.1 29.0 0.1486558 Gent Zomergem 590.1 19.0 0.2268959 Zomergem Peronnes 890.0 55.0 0.659656
10 Voeren Berneau 890.0 5.0 7.2562211 Voeren Berneau 395.5 5.0 0.10803312 Berneau Liege 890.0 20.0 1.8140513 Berneau Liege 395.5 20.0 0.027008414 Liege Warnand 890.0 25.0 1.4512415 Liege Warnand 395.5 25.0 0.021606716 Warnand Namur 890.0 42.0 0.86383617 Namur Anderlues 890.0 40.0 0.90702718 Anderlues Peronnes 890.0 5.0 7.2562219 Peronnes Mons 890.0 10.0 3.6281120 Mons Blaregnies 890.0 25.0 1.4512421 Warnand Wanze 395.5 10.5 0.051444522 Wanze Sinsin 315.5 26.0 0.0064197723 Sinsin Arlon 315.5 98.0 0.0017032024 Arlon Petange 315.5 6.0 0.0278190
Table 3: Pipe-lines description
4.1 Optimal solution
The energy used in the compressors founded by GAMS/CONOPT is:
z∗ = 6 393.825 kW
13
The optimal flow pattern is given in table 4. Note that we have convertedeach couple of two parallel arcs in one equivalent larger pipe (See appendixA).
Arc from to Flow(106 SCM)
1+2 Zeebrugge Dudzele 11.5943+4 Dudzele Brugge 19.994
5 Brugge Zomergem 16.0766 Loenhout Antwerpen 4.8007 Antwerpen Gent 0.7668 Gent Zomergem -4.4909 Zomergem Peronnes 11.586
10 +11 Voeren Berneau 19.34412 +13 Berneau Liege 19.34414+15 Liege Warnand 12.979
16 Warnand Namur 10.83817 Namur Anderlues 8.71818 Anderlues Peronnes 9.91819 Peronnes Mons 22.46420 Mons Blaregnies 15.61621 Warnand Wanze 2.14122 Wanze Sinsin 2.14123 Sinsin Arlon 2.14124 Arlon Petange 1.919
Table 4: Optimal flows
The corresponding pressure and supply optimal patterns are given intable 5. Note that in Voeren, there is one unit of gas that is taken and whichremains at Voeren. This is possible because there is no price associated tothe gas taken at the source nodes.
The two compressors located at Berneau and Sinsin are in use. Table 6give the used powers and the compression ratio pj/pi.
14
Node Town Supply Demand Pressure(106 SCM) (106 SCM) (Bars)
1 Zeebugge 11.594 56.7102 Dudzele 8.400 56.6783 Brugge 3.918 56.5324 Zomergem 54.8695 Loenhout 4.800 56.1746 Antwerpen 4.034 54.0907 Gent 5.256 54.0538 Voeren 20.344 1. 50.0009 in Berneau 49.5799 out Berneau 57.82010 Liege 6.365 56.12611 Warnand 55.14012 Namur 2.120 53.89313 Anderlues 1.200 53.11014 Peronnes 0.960 52.98215 Mons 6.848 51.65316 Blaregnies 15.616 50.00017 Wanze 54.32618 in Sinsin 47.30018 out Sinsin 58.72619 Arlon 0.222 27.52020 Petange 1.919 25.000
Table 5: Optimal supplies and pressures
Localization Wij (kW) Wij (kW) pj/pi pj/pi
Berneau 4 973.991 20 888 1.166 1.6
Sinsin 780.452 3 356 1.242 1.6
Table 6: Compressors energy used
15
4.2 Role of the first problem
Consider now the utility by resorting to the first problem. First of all, wehave asked GAMS/CONOPT to solve directly the complete problem (4) fromscratch. Due to the nonconvexity of the problem, CONOPT can not find afeasible solution to the problem. This fact already justifies the utility of thefirst problem.
But if we consider the solution, in term of flows, of the first problem andof the complete problem, we see that the optimal solution of problem(5) is the optimal solution of problem (4)!
Note that there is some work for CONOPT to solve the complete problem(4) starting from the solution of the first problem (5). In fact, if we considerthe pressure computed from the dual variables of the first problem, we cansee that they are not feasible (See Table 7). As can be seen, the work ofGAMS/CONOPT is to satisfy the upper bound on pressure variables usingthe compressors at a minimum level.
Perhaps this phenomena is due to the arborescent structure of the Belgiangas network. We have try to add arcs on the Belgian gas network toproduce a cycle (See figure 4). We have added two arcs allowing to supplythe city of Arlon directly from the entry point Voeren (distance betweenVoeren and Arlon is 162 km divided in two equal parts of 81 km with adiameter of 395.5 mm (giving a c2
ij = 0.00666873148 for each part). In themiddle point, we have place a compressor with the same maximal power thanat Sinsin, namely 3 356 kW. To force the use of this new canalization, wehave increased all the offers and all the demands by 10% . And the samephenomena appears: the solution, in term of flows of the first problem is theoptimal solution of the complete problem.
This is not so surprizing if we recall the interpretation of the first problem,namely the minimization of the energy dissipated in the network.
5 Conclusions
In this paper we have updated the gas transmission model of De Wolf andSmeers [7] to the new situation in several european countries. Namely thefact that the merchant and the transportation function are now separated.The consequence of this new situation is the necessity of the introduction
16
Node Town Pressure end Pressure end Maximumfirst problem problem (4) pressure
1 Zeebugge 86.664 56.710 77.02 Dudzele 86.634 56.678 77.03 Brugge 86.500 56.532 80.04 Zomergem 84.975 54.869 80.05 Loenhout 86.171 56.174 77.06 Antwerpen 84.263 54.090 80.07 Gent 84.230 54.053 80.08 Voeren 88.019 50.000 50.09 in Berneau 87.686 49.579 66.29 out Berneau 87.686 57.820 66.210 Liege 86.127 56.126 66.211 Warnand 85.223 55.140 66.212 Namur 84.084 53.893 66.213 Anderlues 83.370 53.110 66.214 Peronnes 83.254 52.982 66.215 Mons 82.048 51.653 66.216 Blaregnies 80.555 50.000 66.217 Wanze 84.479 54.326 66.218 in Sinsin 78.139 47.300 63.018 out Sinsin 78.139 58.726 66.219 Arlon 36.505 27.520 66.220 Petange 25.000 25.000 66.2
Table 7: Optimal pressures for first and complete problems
17
Algerian�gas
From storage
Norvegian gas
To France
To Luxemburg
Zeebrugge
Dudzele
BruggeZomergem
Gent
Antwerpen
Loenhout
NodeCompressor
Mons
Blaregnies
PeronnesAnderlues
Warnand-Dreye
Namur
Wanze
Sinsin
Arlon
Berneau
Liege
Voeren
From storage
From storage
Figure 4: Schematic Belgium gas network with a cycle
18
of the compressors modelization. This is the first objective of the presentpaper. The second is to present a procedure to solve this non linear nonconvex non separable problem. We have seen, on the example of the Belgiumgas network that the preprocessing trough the Maugis problem [9] is veryefficient in this case. Namely, on the presented examples, the solution interm of flow variables of the preliminary problem is the optimal solution ofthe gas transmission problem. We have given a physical interpretation tothe objective function of the first problem (5), namely the power used in thenetwork due to the flows in the pipes. This constitutes a justification to thisphenomena.
Futures research are devoted to introduce this model in a dimension-ing model such as [6]. Such a model will consider the trade-off betweenthe minimization of capital expenditures (as in [2]) and the minimization ofoperational expenditures. In other terms, this model could balance any de-crease in investment of pipelines with an increase of compressor power (andconversely) regarding the costs. In fact, if we increase the pipe diameters,this leads to minus head loses and perhaps the use of one compressor can beavoided.
Acknowledgment
Thanks are due to Mr Jean Andre, of “Direction de la Recherche de Gazde France” which provided most of the funding for the modelization of thecompressors. Our gratitude also extends to Mr Zarea of Gaz de France forhelping us to find a physical interpretation to the first problem.
19
A Formula for two parallel pipes-lines
Consider two parallel pipes-lines (indiced by a and b) between nodes i and j.We note respectively fa
ij and f bij the flow in the two parallel arcs. The (4.2)
formula can be written:
sign(faij)(f
aij)
2 = (Caij)
2(p2i − p2
j)
sign(f bij)(f
bij)
2 = (Cbij)
2(p2i − p2
j)
Since, at the optimum, the flows in the two parallel arcs must be in thesame direction, we can choose the direction of arc (i, j) so that the two signfunctions are equal to one:
(faij)
2 = (Caij)
2(p2i − p2
j)
(f bij)
2 = (Cbij)
2(p2i − p2
j)
If we replace the two parallel arcs by one equivalent greater diameter withcorresponding parameter Cij, the corresponding flow fij is the sum of thetwo flows:
fij = faij + f b
ij
We compute now the relation between, on one side, Cij and, on the otherside, Ca
ij and Cbij:
f 2ij = (fa
ij + f bij)
2
= (faij)
2 + 2faijf
bij + (f b
ij)2
= (Caij)
2(p2i − p2
j) + 2Caij
√(p2
i − p2j)C
bij
√(p2
i − p2j) + (Cb
ij)2(p2
i − p2j)
= [(Caij)
2 + 2CaijC
bij + (Cb
ij)2](p2
i − p2j)
The (4.2) formula can be written:
f 2ij = C2
ij[p2i − p2
j ]
By identification of the two last equations we deduced the desired formula:
C2ij = (Ca
ij)2 + 2Ca
ijCbij + (Cb
ij)2 (6)
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References
[1] Jean ANDRE, Note sur les modeles simplifies de stations de compres-sions, privated communication, Direction de la recherche de Gaz deFrance, April 2006.
[2] Jean ANDRE, Frederic BONNANS and Laurent CORNIBERT, Plan-ning reinforcement of gas transportation networks with optimizationmethods, Proceedings of the 19th Mini EURO Conference (ORMMES06), 6-9 September 2006
[3] Jean ANDRE, Laurent CORNIBERT, A tool to optimize the Reinforce-ment Costs of a Gas Transmission Network, Pipeline Simulation InterestGroup, 38th Annual Meeting, Williamsburg (VA), October 2006.
[4] BERTSEKAS, D. 1995, Nonlinear Programming, Athena Scientific, Bel-mont.
[5] B.V. BABU, Rakesh ANGIRA, Pallavi G. CHARKOLE and J. H. SyedMUBEEN, Optimal Design of Gas Transmission Network, Using Differ-ential Evolution, Department of Chemical Engineering, Birla Instituteof Technology Science, Pilani, India, 2003.
[6] Daniel DE WOLF and Yves SMEERS, Optimal Dimensioning of PipeNetworks with Application to Gas Transmission Networks, OperationsResearch,Vol 44, No 4, July-August 1996, pp 596-608.
[7] Daniel DE WOLF and Yves SMEERS, The Gas Transmission Prob-lem Solved by an Extension of the Simplex Algorithm, ManagementSciences, Vol. 46, No 11, November 2000, pp 1454-1465
[8] Daniel DE WOLF, Mathematical Properties of Formulations of the GasTransmission Problem, submitted april 2004 to RAIRO in the field ”Op-erations Research”.
[9] MAUGIS, M.J.J. 1977, Etude de Reseaux de Transport et de Distribu-tion de Fluides, R.A.I.R.O., Vol.11, No 2, pp 243-248.
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[10] Seugwon AN, Qing LI, Thomas W. GEDRA, Natural Gas and Elec-tricity Optimal Power Flow, Oklahoman State University, IEEE Paper,2003.
[11] O’NEILL, R.P., WILLIARD, M., WILKINS, B. and R. PIKE 1979,A Mathematical Programming Model for Allocation Of Natural Gas,Operations Research 27, No 5, pp 857-873.
[12] WILSON, J.G., WALLACE, J. and B.P. FUREY 1988, Steady-stateOptimization of Large gas transmission systems, in Simulation and op-timization of large systems, A.J. Osiadacz Ed, Clarendon Press, Oxford.
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